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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.06.0056.01

 Input Form

 Hypergeometric2F1[a, b, c, z] == Subscript[F, Infinity][z, a, b, c] /; (Subscript[F, n][z, a, b, c] == (((Gamma[b - a] Gamma[c])/(Gamma[b] Gamma[c - a])) Sum[(Pochhammer[a, k] Pochhammer[a - c + 1, k])/ (k! Pochhammer[a - b + 1, k])/z^k, {k, 0, n}])/(-z)^a + (((Gamma[a - b] Gamma[c])/(Gamma[a] Gamma[c - b])) Sum[(Pochhammer[b, k] Pochhammer[b - c + 1, k])/ (k! Pochhammer[b - a + 1, k])/z^k, {k, 0, n}])/(-z)^b == Hypergeometric2F1[a, b, c, z] - (Gamma[-a + b] Gamma[c] Pochhammer[a, 1 + n] (Pochhammer[1 + a - c, 1 + n]/ ((1 + n)! Gamma[b] Gamma[-a + c] Pochhammer[1 + a - b, 1 + n])) z^(-n - 1) HypergeometricPFQ[{1, 1 + a + n, 2 + a - c + n}, {2 + n, 2 + a - b + n}, 1/z])/(-z)^a - (((Gamma[a - b] Gamma[c] Pochhammer[b, 1 + n] Pochhammer[1 + b - c, 1 + n])/((1 + n)! Gamma[a] Gamma[-b + c] Pochhammer[1 - a + b, 1 + n])) z^(-n - 1) HypergeometricPFQ[{1, 1 + b + n, 2 + b - c + n}, {2 + n, 2 - a + b + n}, 1/z])/(-z)^b && Element[n, Integers] && n >= 0) && !Element[a - b, Integers]

 Standard Form

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 MathML Form

 2 F 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] F ( z , a , b , c ) /; ( ( F n ( z , a , b , c ) Γ ( b - a ) Γ ( c ) ( - z ) - a Γ ( b ) Γ ( c - a ) k = 0 n ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( a - c + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "c", "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( a - b + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a - b ) Γ ( c ) ( - z ) - b Γ ( a ) Γ ( c - b ) k = 0 n ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] ( b - c + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a + b + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", "1"]], ")"]], "k"], Pochhammer] 2 F 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - Γ ( b - a ) Γ ( c ) ( a ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( a - c + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "c", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] z - n - 1 ( n + 1 ) ! Γ ( b ) Γ ( c - a ) ( a - b + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( - z ) - a 3 F 2 ( 1 , a + n + 1 , a - c + n + 2 ; n + 2 , a - b + n + 2 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "c", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "b", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - Γ ( a - b ) Γ ( c ) ( b ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( b - c + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] z - n - 1 ( n + 1 ) ! Γ ( a ) Γ ( c - b ) ( - a + b + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( - z ) - b 3 F 2 ( 1 , b + n + 1 , b - c + n + 2 ; n + 2 , - a + b + n + 2 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) ¬ a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric2F1 a b c z Subscript F z a b c Subscript F n z a b c Gamma b -1 a Gamma c -1 z -1 a Gamma b Gamma c -1 a -1 k 0 n Pochhammer a k Pochhammer a -1 c 1 k z -1 k k Pochhammer a -1 b 1 k -1 Gamma a -1 b Gamma c -1 z -1 b Gamma a Gamma c -1 b -1 k 0 n Pochhammer b k Pochhammer b -1 c 1 k z -1 k k Pochhammer -1 a b 1 k -1 Hypergeometric2F1 a b c z -1 Gamma b -1 a Gamma c Pochhammer a n 1 Pochhammer a -1 c 1 n 1 z -1 n -1 n 1 Gamma b Gamma c -1 a Pochhammer a -1 b 1 n 1 -1 -1 z -1 a HypergeometricPFQ 1 a n 1 a -1 c n 2 n 2 a -1 b n 2 1 z -1 -1 Gamma a -1 b Gamma c Pochhammer b n 1 Pochhammer b -1 c 1 n 1 z -1 n -1 n 1 Gamma a Gamma c -1 b Pochhammer -1 a b 1 n 1 -1 -1 z -1 b HypergeometricPFQ 1 b n 1 b -1 c n 2 n 2 -1 a b n 2 1 z -1 n a -1 b [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2007-05-02